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15
On the Notion of Interestingness in Automated Mathematical Discovery
 International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical ..."
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Cited by 64 (25 self)
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We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
Agent based cooperative theory formation in pure mathematics
 In Proceedings of the AISB00 Symposium on Creative & Cultural Aspects and Applications of AI & Cognitive Science
, 2000
"... The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multiagent version of HR which may ..."
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Cited by 12 (7 self)
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The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multiagent version of HR which may provide a model for how individual mathematicians perform separate investigations but communicate their results to the mathematical community, learning from others as they do. We detail the exhaustive categorisation problem to which we have applied a multiagent approach. 1
Letter Spirit: An Emergent Model of the Perception and Creation of Alphabetic Style
 Center for
, 1993
"... The Letter Spirit project is an attempt to model central aspects of human highlevel perception and creativity on a computer, focusing on the creative act of artistic letterdesign. The aim is to model the process of rendering the 26 lowercase letters of the roman alphabet in many different, interna ..."
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Cited by 9 (2 self)
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The Letter Spirit project is an attempt to model central aspects of human highlevel perception and creativity on a computer, focusing on the creative act of artistic letterdesign. The aim is to model the process of rendering the 26 lowercase letters of the roman alphabet in many different, internally coherent styles. Two important and orthogonal aspects of letterforms are basic to the project: the categorical sameness possessed by instances of a single letter in various styles (e.g., the letter `a' in Baskerville, Palatino, and Helvetica) and the stylistic sameness possessed by instances of various letters in a single style (e.g., the letters `a', `b', and `c' in Baskerville). Starting with one or more seed letters representing the beginnings of a style, the program will attempt to create the rest of the alphabet in such a way that all 26 letters share the same style, or spirit. Letters in the domain are formed exclusively from straight segments on a grid in order to make decisions ...
Cross domain mathematical concept formation
 In Proceedings of the AISB00 Symposium on Creative & Cultural Aspects and Applications of AI & Cognitive Science
, 2000
"... Many interesting concepts in mathematics are essentially ‘crossdomain ’ in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an inves ..."
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Cited by 5 (1 self)
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Many interesting concepts in mathematics are essentially ‘crossdomain ’ in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an investigation seems to exercise a mathematician’s creative ability. The HR program, (Colton et al., 1999), proposes new concepts in mathematics. Its original implementation was limited to working in one mathematical domain at a time, so it was unable to create crossdomain concepts. Here, we describe an extension of HR to multiple domains. Crossdomain concept formation is facilitated by generalisation of the data structures and heuristic measures employed by the program, and the implementation of a new production rule. Results achieved include generation of the concepts of prime order groups, graph nodes of maximal degree and an interesting class of graph. 1
Automatic Theory Formation in Graph Theory
 In Argentine Symposium on Artificial Intelligence
, 1999
"... This paper presents SCOT, a system for automatic theory construction in the domain of Graph Theory. Following on the footsteps of the programs ARE [9], HR [1] and Cyrano [6], concept discovery is modeled as search in a concept space. We propose a classification for discovery heuristics, which takes ..."
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Cited by 4 (0 self)
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This paper presents SCOT, a system for automatic theory construction in the domain of Graph Theory. Following on the footsteps of the programs ARE [9], HR [1] and Cyrano [6], concept discovery is modeled as search in a concept space. We propose a classification for discovery heuristics, which takes into account the main processes related to theory construction: concept construction, example production, example analysis, conjecture construction, and conjecture analysis.
Knowledgelevel creativity in game design
 In Proc. of the 2nd International Conference in Computational Creativity (ICCC
, 2011
"... Drawing on inspirations outside of traditional computational creativity domains, we describe a theoretical explanation of creativity in game design as a knowledge seeking process. This process, based on the practices of human game designers and an extended analogy with creativity in science, is amen ..."
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Cited by 4 (3 self)
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Drawing on inspirations outside of traditional computational creativity domains, we describe a theoretical explanation of creativity in game design as a knowledge seeking process. This process, based on the practices of human game designers and an extended analogy with creativity in science, is amenable to computational realization in the form of a discovery system. Further, the model of creativity it entails, creativity as the rational pursuit of curiosity, suggests a new perspective on existing artifact generation challenges and prompts a new mode of evaluation for creative agents (both human and machine).
Understanding complex systems through examples: A framework for qualitative example finding
 Kingston University
, 2000
"... Many complex systems have the characteristic that we can classify objects in the system in some way, but that these classi cations are distributed through a parameter space in some complex fashion. In order for a human to get an understanding of the system, we would like to present this user with on ..."
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Cited by 3 (2 self)
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Many complex systems have the characteristic that we can classify objects in the system in some way, but that these classi cations are distributed through a parameter space in some complex fashion. In order for a human to get an understanding of the system, we would like to present this user with one example of an object for each class. Examples of such problems can be found in information retrieval, bioinformatics, computational geometry, computeraided design, software testing and cellular automata. In this paper we will show how problems in all these areas can be put into a general framework of nding qualitative examples, and argue that general heuristic approaches to this type of problem are an important and neglected area of machine learning. We contrast this with some other wellstudied problems, showing how this problem is distinct and investigating what we can learn from these problems. We then discuss some of the requirements for a heuristic to solve these problems,...
Mathematical Applications of Inductive Logic Programming
"... Abstract. The application of Inductive Logic Programming to scientific datasets has been highly successful. Such applications have led to breakthroughs in the domain of interest and have driven the development of ILP systems. The application of AI techniques to mathematical discovery tasks, however, ..."
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Cited by 2 (2 self)
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Abstract. The application of Inductive Logic Programming to scientific datasets has been highly successful. Such applications have led to breakthroughs in the domain of interest and have driven the development of ILP systems. The application of AI techniques to mathematical discovery tasks, however, has largely involved computer algebra systems and theorem provers rather than machine learning systems. We discuss here the application of the HR and Progol machine learning programs to discovery tasks in mathematics. While Progol is an established ILP system, HR has historically not been described as an ILP system. However, many applications of HR have required the production of first order hypotheses given data expressed in a Prologstyle manner, and the core functionality of HR can be expressed in ILP terminology. In (Colton, 2003), we presented the first partial description of HR as an ILP system, and we build on this work to provide a full description here. HR performs a novel ILP routine called Automated Theory Formation, which combines inductive and deductive reasoning to form clausal theories consisting of classification rules and association rules. HR generates definitions using a set of production rules,
Crossdomain Mathematical Concept Formation
 In Proceedings of the AISB00 Symposium on Creative & Cultural Aspects and Applications of AI & Cognitive Science
, 2000
"... Many interesting concepts in mathematics are essentially `crossdomain' in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an inve ..."
Abstract
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Many interesting concepts in mathematics are essentially `crossdomain' in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an investigation seems to exercise a mathematician's creative ability. The HR program, (Colton et al., 1999), proposes new concepts in mathematics. Its original implementation was limited to working in one mathematical domain at a time, so it was unable to create crossdomain concepts. Here, we describe an extension of HR to multiple domains. Crossdomain concept formation is facilitated by generalisation of the data structures and heuristic measures employed by the program, and the implementation of a new production rule. Results achieved include generation of the concepts of prime order groups, graph nodes of maximal degree and an interesting class of graph. 1 Introduction In previous wor...
An Attempt to Automate NPHardness Reductions via SO∃ Logic
, 2004
"... We explore the possibility of automating NPhardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of secondorder existential (SO#) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio ..."
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We explore the possibility of automating NPhardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of secondorder existential (SO#) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio Medina, we explore the possibility of implementing seven syntactic operators. Each operator transforms SO# sentences in a way that preserves NPcompleteness. We subsequently propose a program which implements these operators.